but unlike Ash Wednesday it's not an obligation
Ash Wednesday is not a day of obligation.
The primary purpose of a penitential service for Lent (or Advent as well) is confession. Everything in addition to that is to aid in making a good confession and encouraging people to go
- Is there a document from Vatican governing the format, or can it vary depending on country / diocese? If not, what are the necessary elements?
There is the Rite of Penance, a book published in 1973, which describes the Sacrament of Confession as it is to be ordinarily practiced in the Roman Rite. Chapter IV-B details the rite for multiple penitents each receiving individual absolution. It can vary in accordance to the existing norms on variance from the Editio Typica of a given ritual. For example at a parish run by a Latin Rite diocese, they would use this rite. At a parish run by an Ecclesia Dei community, they would instead use the 1952 Rituale Romanum.
- What is the sacramental nature of this service?
The Sacrament itself is the confession and absolution. If the priest does not say "I absolve you" (Ego Te Absolvo), there is no Sacrament. Everything aside from that is a sacramental. Which is, in short, a thing which is a source of actual grace subject to the Sanctifying power of the Church (some of which remit venial sins). The video you linked depicts no Sacrament. As the description says, individual confessions followed.
- Is the text of the prayers in the service standard? Is the choice of Bible reading correspond to the assigned reading of the day, specially standardized for this service, or at the bishop/priest's discretion?
Readings should be chosen which illustrate the following:
a) the voice of God calling men back to conversion and ever closer conformity with Christ;
b) the mystery of our reconciliation through the death and resurrection of Christ and through the gift
of the Holy Spirit;
c) the judgment of God about good and evil in men’s lives as a help in the examination of conscience
So, officially it's at the discretion of the presiding cleric (or more likely, whoever is the "liturgical planner"). But technically he ought to in good faith pick a passage fitting the above criteria.
- Why is it separated from Ash Wednesday? What was its origin?
As to the latter, I'm not sure of it originating sooner than the publication of the Novus Ordo Missae. Prior to this it seems people would simply go stand in line for confession, and maybe sometimes there'd be special occasions where a bunch of priests went to one Church and advertised it so people would be encouraged to go.
Ash Wednesday is Ash Wednesday and penance services are penance services. Ash Wednesday is quite ancient, appearing in the Gregorian Sacramentary around the 8th Century. There's not really a relation between the two aside from their place in Lent and their penitential character.
- What is the purpose behind this service?
To encourage people to go to Confession, and to make a good confession.
- Does a parish offer it more than one time per year? Can it be conducted upon parishioners's request?
It happens during Advent as well for similar reasons as in Lent. In some places a diocese will coordinate multiple services in the season, spreading them so that everyone will have at least one within reasonable distance at least once during the season. The service is meant for a large group of people, so it doesn't make much sense for just one person to request it for themselves. They can simply go to confession at the usual times.